[FOM] Higher Order Set Theory

[Roger Bishop Jones]

On Wednesday 09 March 2005 11:26 am, Aatu Koskensilta wrote:

> On Mar 8, 2005, at 10:57 AM, Roger Bishop Jones wrote:

> > Higher order set theory is a great deal less problematic
> > than you make it appear.
>
> In a sense higher order set theory is not problematic at all
> and has a natural axiomatization in Morse-Kelley set theory.
> However, from a conceptual point of view, I believe that
> higher order set theory is *much* more problematic than the
> universe of sets, V alone. The problem is that it's not
> obvious how there could be a determined totality of
> subcollections of V over which the second order quantifiers
> were to range over.

Its not at all obvious to me that the supposition that V
exists is coherent.
Obvious proofs to the contrary spring to mind.

How can something be more problematic than an incoherent
supposition?

The only problem here is with V, the supposition of which
is necessary neither for first order nor for higher order
set theory.

Roger Jones